Equilateral Triangle Formula: Overview, Properties, Examples

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Equilateral Triangle Formula: Overview

An equilateral triangle is a closed two-dimensional figure with three equal-length sides and three corners.

All the sides of an equilateral triangle are equal, and all the interior angles are \(60^{\circ}\). As a result, if one side’s length is known, the area of an equilateral triangle may be computed.

Basic Properties of an Equilateral Triangle

Some of the basic properties of an equilateral triangle are:

An equilateral triangle is a regular three-sided polygon.
Each of the three sides in an equilateral triangle is equal.
All three angles of an equilateral triangle are equal to \(60^{\circ}\) and are congruent.
The perpendicular drawn from the equilateral triangle’s vertex to the opposite side divides it in half. In addition, the vertex from which the perpendicular is drawn is divided into two equal angles, each of which is \(30^{\circ}\).
Both the ortho-centre and the centroid are located at the same location.
All sides of an equilateral triangle have the same median, angle bisector, and altitude.

Also, Check:

Triangles	Properties of Triangles
Area of Triangle	Area of Equilateral Triangle
Area of Right Angled Triangle	Geometry Formulae

Equilateral Triangle Formula to Find its Perimeter

The perimeter of an equilateral triangle is calculated by summing the lengths of all three sides.

Perimeter of equilateral triangle’s \(=\) Sum of the three sides’ lengths

Perimeter of an equilateral triangle is given by \(=b+b+b=3 b \,\text {units}\)

Formula for Area of an Equilateral Triangle

One should know the length of its side to find the area of an equilateral triangle.

The area of an equilateral triangle is given by,

\(A=\frac{\sqrt{3}}{4} \times(\text {side})^{2} \,\text {square units}\)

Equilateral Triangle Formula Derivation

The line segment from a vertex that is perpendicular to the opposite side is the altitude or height of an equilateral triangle. An equilateral triangle’s altitude bisects both its base and the opposite angle.

Here \(h=\,\text {height}\), \(a=\,\text {side length}\)

Apply the Pythagoras theorem for the \(\triangle A B D\),\(B D^{2}+A D^{2}=A B^{2} \ldots…(i)\)\(\Rightarrow\left(\frac{a}{2}\right)^{2}+h^{2}=a^{2}\)\(\Rightarrow h^{2}=a^{2}-\left(\frac{a}{2}\right)^{2}\)\(\Rightarrow h^{2}=a^{2}-\frac{a^{2}}{2^{2}}\)\(\Rightarrow h^{2}=\frac{4 a^{2}-a^{2}}{4}\)\(\Rightarrow h^{2}=\frac{3 a^{2}}{4}\)\(\Rightarrow h=\sqrt{\frac{3 a^{2}}{4}}\)\(\Rightarrow h=\frac{\sqrt{3} \times a}{2}\)\(\Rightarrow h=\frac{a \sqrt{3}}{2} \quad \ldots..(ii)\)Area of the triangle \(A=\frac{1}{2} \times \text {base} \times \text {height}\)\(\Rightarrow A=\frac{1}{2} \times b \times h \ldots..(iii)\)

Substitute equation \((ii)\) in equation \((iii)\).

Here, base \(b=\frac{a}{2}+\frac{a}{2}=\frac{2 a}{2}=a\)\(A=\frac{1}{2} \times a \times \frac{a \sqrt{3}}{2}\)\(A=\frac{\sqrt{3} \times a^{2}}{4}\)\(A=\frac{\sqrt{3}}{4} \times a^{2}\)

Hence, it is proved.

Learn All the Concepts on Scalene Triangle

Equilateral Triangle Formula to Find Area: Heron’s Formula

Heron’s formula or Hero’s formula can be used to derive a special formula applicable to calculate the area of an equilateral triangle only.

In an equilateral triangle, the lengths of all the sides are the same. So, \(a=b=c\).

Therefore, \(s=\frac{a+b+c}{2}=\frac{a+a+a}{2}=\frac{3 a}{2}\)

So, the area \(=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{\frac{3 a}{2} \times\left(\frac{3 a}{2}-a\right) \times\left(\frac{3 a}{2}-a\right) \times\left(\frac{3 a}{2}-a\right)}\)

\(=\sqrt{\frac{3 a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}}=\sqrt{\frac{3 a^{4}}{16}}=\frac{\sqrt{3}}{4} a^{2}\), where the length of the side of the triangle is \(a\).

Therefore, the area of an equilateral triangle is \(=\frac{\sqrt{3}}{4} \times a^{2}=\frac{\sqrt{3}}{4} \times(\text {side})^{2}\).

Equilateral Triangle Formula to Find Length

Case – 1: We can find the length of an equilateral triangle if the perimeter is given,\(\text {Length of a side} =\frac{\text { perimeter }}{3}\)

Case-2: We can find the length of an equilateral triangle if the area is given,\({\text{Length}}\,{\text{of}}\,{\text{a}}\,{\text{side}} = \frac{{2\sqrt {{\text{ area}}} }}{3}\)

Equilateral Triangle Formula to Find Height/Altitude

The line segment from a vertex perpendicular to the opposite side is the altitude or height of an equilateral triangle. An equilateral triangle’s altitude bisects both its base and the opposite angle.

If you draw a height/altitude in an equilateral triangle, we can see that the triangle is divided into two right-angled triangles in which: sides \(a\) are hypotenuses, heigh/altitude is common for both triangles, the other side is equal to \(\frac{a}{2}\), therefore we can use the Pythagorean theorem.

\(\left(\frac{a}{2}\right)^{2}+h^{2}=a^{2}\)\(\Rightarrow h^{2}=a^{2}-\left(\frac{a}{2}\right)^{2}\)\(\Rightarrow h^{2}=a^{2}-\frac{a^{2}}{2^{2}}\)\(\Rightarrow h^{2}=\frac{4 a^{2}-\alpha^{2}}{4}\)\(\Rightarrow h^{2}=\frac{3 a^{2}}{4}\)\(\Rightarrow h=\sqrt{\frac{3 a^{2}}{4}}\)\(\Rightarrow h=\frac{\sqrt{3} \times a}{2}\)\(\Rightarrow h=\frac{a \sqrt{3}}{2}\)

Hence, \(h=\frac{a \sqrt{3}}{2}\) is the formula to find the height/altitude of an equilateral triangle.

Solved Examples on Equilateral Triangle Formula

Q.1. Find the area of an equilateral triangle of side \(5 \mathrm{~cm}\).Ans: Here, \(a=5 \mathrm{~cm}\)

Formula to find the area of an equilateral triangle is \(=\frac{\sqrt{3}}{4} \times a^{2}\)\(\Rightarrow \frac{\sqrt{3}}{4} \times(5)^{2}\)\(\Rightarrow \frac{\sqrt{3}}{4} \times 25\)\(\Rightarrow \frac{25 \sqrt{3}}{4} \mathrm{~cm}^{2}\)Hence, the required area of an equilateral triangle with a side \(5 \mathrm{~cm}\) is \(\frac{25 \sqrt{3}}{4} \mathrm{~cm}^{2}\).

Q.2. Find the area of an equilateral triangle whose side is \(6 \mathrm{~cm}\).Ans: Formula to find the area of an equilateral triangle is \(=\frac{\sqrt{3}}{4} \times a^{2}\)\(=\frac{\sqrt{3}}{4} \times(6)^{2}\)\(=\frac{\sqrt{3}}{4} \times 36\)\(=9 \sqrt{3} \mathrm{~cm}^{2}\)Hence, the obtained area of an equilateral triangle with a side \(6 \mathrm{~cm}\) is \(9 \sqrt{3} \mathrm{~cm}^{2}\).

Q.3. Calculate the area of an equilateral triangle whose each side is \(12 \mathrm{~cm}\) ?Ans: Formula to find the area of an equilateral triangle is \(=\frac{\sqrt{3}}{4} \times a^{2}\)\(\Rightarrow \frac{\sqrt{3}}{4} \times(12)^{2}\)\(\Rightarrow \frac{\sqrt{3}}{4} \times 144\)\(\Rightarrow 36 \sqrt{3} \mathrm{~cm}^{2}\)Hence, the obtained area of an equilateral triangle with a side \(12 \mathrm{~cm}\) is \(36 \sqrt{3} \mathrm{~cm}^{2}\).

Q.4. Find the perimeter of the equilateral triangle with the side length \(6.5 \mathrm{~cm}\).Ans: From the given, \(\text {side length} =6.5 \mathrm{~cm}\)Perimeter of an equilateral triangle is given by \(=b+b+b=3 b \,\text {units}\), where \(b=\text {side length}\)\(\Rightarrow 6.5+6.5+6.5=19.5 \mathrm{~cm}\)Hence, the obtained perimeter of an equilateral triangle with a side \(6.5 \mathrm{~cm}\) is \(19.5 \mathrm{~cm}\).

Q.5. What is the height of an equilateral triangle with a side length of \(8 \mathrm{~cm}\) ?Ans: From the given, \(\text {side length}\,\, a=8 \mathrm{~cm}\)Formula to find an equilateral triangle is given by,\(\Rightarrow h=\frac{a \sqrt{3}}{2}\)\(\Rightarrow h=\frac{\sqrt{3}}{2} \times 8\)\(\Rightarrow h=4 \sqrt{3} \mathrm{~cm}\)Hence, \(4 \sqrt{3} \mathrm{~cm}\) is the height of an equilateral triangle with a side length of \(8 \mathrm{~cm}\).

Q.6. Find the side length of an equilateral triangle whose area is \(9 \mathrm{~cm}^{2}\).Ans: From the given, Area \(=9 \mathrm{~cm}^{2}\)\(\text {Side length}=\frac{2 \sqrt{\text { area }}}{3}\)\(\Rightarrow\text {Side length}=\frac{2 \sqrt{9}}{3}\)\(\Rightarrow\text {Side length}=\frac{2 \times 3}{3}\)\(\Rightarrow\text {Side length}=2 \mathrm{~cm}\)Hence, the \(2 \mathrm{~cm}\) is the side length of an equilateral triangle whose area is \(9 \mathrm{~cm}^{2}\).

Q.7. Find the side length of an equilateral triangle whose area is \(16 \mathrm{~cm}^{2}\).Ans: From the given, Area \(=16 \mathrm{~cm}^{2}\)\(\text {Side length} =\frac{2 \sqrt{\text { area }}}{3}\)\(\Rightarrow\text {Side length}=\frac{2 \sqrt{16}}{3}\)\(\Rightarrow\text {Side length}=\frac{2 \times 4}{3}\)\(\Rightarrow\text {Side length}=\frac{8}{3} \mathrm{~cm}\)Hence, the \(\frac{8}{3} \mathrm{~cm}\) is the side length of an equilateral triangle whose area is \(16 \mathrm{~cm}^{2}\).

Q.8. Find the perimeter of the equilateral triangle with the side length \(8 \mathrm{~cm}\).Ans: Given that the \(\text {side length} =8 \mathrm{~cm}\)Perimeter of an equilateral triangle is given by \(=b+b+b=3b \,\text {units}\), where \(b=\,\text {side length}\)\(\Rightarrow 8+8+8=24 \mathrm{~cm}\)Hence, the obtained perimeter of an equilateral triangle with a side \(8 \mathrm{~cm}\) is \(24 \mathrm{~cm}\).

Summary

An equilateral triangle has three equal sides, and all of its internal angles are \(60\) degrees. As a result, the area of an equilateral triangle may be calculated if one side’s length is known. The purpose of this article is to teach you how to calculate the perimeter, area, and height of an equilateral triangle based on the information supplied.

Perimeter of an equilateral triangle is given by \(=b+b+b=3 b \,\text {units}\). The equilateral triangle formula for finding height is \(h=\frac{a \sqrt{3}}{2}\). The area of an equilateral triangle is \(A=\frac{\sqrt{3}}{4} \times(\text {side})^{2} \,\text {square units}\).

FAQs on Equilateral Triangle Formula

Q.1. What is the formula of an equilateral triangle?Ans: Formula to find an area of an equilateral triangle is given by,\(A=\frac{\sqrt{3}}{4} \times(\text {side})^{2} \,\text {square units}\)And, formula to find the perimeter of an equilateral triangle is given by,\(P=c+c+c=3 c\,\text {units}\), where side \(=c\) units.

Q.2. What is the formula of equilateral triangle height?Ans: Formula to find the height of an equilateral triangle is given by,\(\Rightarrow\text {height} = \text {side} \times \frac{\sqrt{3}}{2} \text {units}\)

Q.3. How do you find the length of one side of an equilateral triangle?Ans: Case -1: We can find the length of an equilateral triangle if the perimeter is given,\(\text {Length of a side} =\frac{\text { perimeter }}{3}\)Case-2: We can find the length of an equilateral triangle if the area is given,\({\text{Length}}\,{\text{of}}\,{\text{a}}\,{\text{side}} = \frac{{2\sqrt {{\text{ area}}} }}{3}\)

Q.4. How to calculate the equilateral triangle height formula?Ans: If we divide the equilateral triangle into two equal parts and give the values \(p, q\) and \(r\) Consider the hypotenuse as \(r\) and side \(p\) will be equal to half of the side length, and side \(q\) is the height of the equilateral triangle. Then, by using the Pythagoras theorem, we can find the height of an equilateral triangle.\(\Rightarrow q=\frac{p \sqrt{3}}{2}\)where, \(q=\text {height}\), \(p=\text {side length}\)

Q.5. What is the area of an equilateral triangle formula when height is given.Ans: The formula to find the area of an equilateral triangle when the height is given is,\(h=\frac{(a \times \sqrt{3})}{2}\)\(\Rightarrow a=\frac{2 h}{\sqrt{3}}\)Where \(h=\text {height}\), \(a=\text {side}\)

Q.6. What are the properties of an equilateral triangle?Ans: Properties of an equilateral triangle are,1. An equilateral triangle is a regular three-sided polygon.2. Each of the three sides in an equilateral triangle is equal.3. All three angles of an equilateral triangle are equal to \(60^{\circ}\) and are congruent.4. The perpendicular drawn from the equilateral triangle’s vertex to the opposite side divides it in half. In addition, the vertex from which the perpendicular is drawn is divided into two equal angles, each of which is \(30^{\circ}\).5. Both the ortho-centre and the centroid are located at the same location.6. All sides of an equilateral triangle have the same median, angle bisector, and altitude.

Q.7. What is the height of an equilateral triangle?Ans: An equilateral triangle has three congruent sides and is also an equiangular triangle, which has three congruent angles, each of which is \(60^{\circ}\). By drawing a line from one corner to the centre of the opposite side, we can divide the triangle into two exceptional \(30 – 60 – 90\) right triangles.

We hope this detailed article on equilateral triangle formulas helped you in your studies. If you have any doubts or queries regarding this topic, feel to ask us in the comment section. Happy learning!

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